{"title":"Degeneration of 7-Dimensional Minimal Hypersurfaces Which are Stable or Have a Bounded Index","authors":"Nick Edelen","doi":"10.1007/s00205-024-02003-w","DOIUrl":null,"url":null,"abstract":"<div><p>A 7-dimensional area-minimizing embedded hypersurface <span>\\(M^7\\)</span> will in general have a discrete singular set, and the same is true if <i>M</i> is locally stable provided <span>\\({\\mathcal {H}}^6(\\textrm{sing}M) = 0\\)</span>. We show that if <span>\\(M_i^7\\)</span> is a sequence of 7D minimal hypersurfaces which are minimizing, stable, or have bounded index, then <span>\\(M_i \\rightarrow M\\)</span> can limit to a singular <span>\\(M^7\\)</span> with only very controlled geometry, topology, and singular set. We show that one can always “parameterize” a subsequence <span>\\(i'\\)</span> with controlled bi-Lipschitz maps <span>\\(\\phi _{i'}\\)</span> taking <span>\\(\\phi _{i'}(M_{1'}) = M_{i'}\\)</span>. As a consequence, we prove the space of smooth, closed, embedded minimal hypersurfaces <i>M</i> in a closed Riemannian 8-manifold <span>\\((N^8, g)\\)</span> with a priori bounds <span>\\({\\mathcal {H}}^7(M) \\leqq \\Lambda \\)</span> and <span>\\(\\textrm{index}(M) \\leqq I\\)</span> divides into finitely-many diffeomorphism types, and this finiteness continues to hold if one allows the metric <i>g</i> to vary, or <i>M</i> to be singular.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6000,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02003-w.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Rational Mechanics and Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00205-024-02003-w","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
A 7-dimensional area-minimizing embedded hypersurface \(M^7\) will in general have a discrete singular set, and the same is true if M is locally stable provided \({\mathcal {H}}^6(\textrm{sing}M) = 0\). We show that if \(M_i^7\) is a sequence of 7D minimal hypersurfaces which are minimizing, stable, or have bounded index, then \(M_i \rightarrow M\) can limit to a singular \(M^7\) with only very controlled geometry, topology, and singular set. We show that one can always “parameterize” a subsequence \(i'\) with controlled bi-Lipschitz maps \(\phi _{i'}\) taking \(\phi _{i'}(M_{1'}) = M_{i'}\). As a consequence, we prove the space of smooth, closed, embedded minimal hypersurfaces M in a closed Riemannian 8-manifold \((N^8, g)\) with a priori bounds \({\mathcal {H}}^7(M) \leqq \Lambda \) and \(\textrm{index}(M) \leqq I\) divides into finitely-many diffeomorphism types, and this finiteness continues to hold if one allows the metric g to vary, or M to be singular.
一个 7 维面积最小的内嵌超曲面 \(M^7\) 一般会有一个离散奇异集,如果 M 是局部稳定的,只要 \({\mathcal {H}}^6(\textrm{sing}M) = 0\) 也是如此。我们证明,如果\(M_i^7\)是一个最小化、稳定或有界索引的7D最小超曲面序列,那么\(M_i \rightarrow M\) 可以极限到一个奇异的\(M^7\),其几何、拓扑和奇异集都非常受控。我们证明,我们总是可以用受控的双立普茨映射 \(\phi _{i'}\) 取 \(\phi _{i'}(M_{1'}) = M_{i'}\ 来 "参数化 "子序列 \(i'\)。因此,我们证明了在封闭的黎曼 8-manifold((N^8、g)\) with a priori bounds \({\mathcal {H}}^7(M) \leqq \Lambda \) and \(\textrm{index}(M) \leqq I\) divides into finitely-many diffomorphism types, and this finiteness continues to hold if one allows the metric g to vary, or M to be singular.
期刊介绍:
The Archive for Rational Mechanics and Analysis nourishes the discipline of mechanics as a deductive, mathematical science in the classical tradition and promotes analysis, particularly in the context of application. Its purpose is to give rapid and full publication to research of exceptional moment, depth and permanence.